Collins Fragmentation Function within NJL

Collins fragmentation function within NJL-jet model
Hrayr H. Matevosyan,1 Anthony W. Thomas,1 and Wolfgang Bentz2
arXiv:1205.5813v2 [hep-ph] 28 Aug 2012
1
CSSM and ARC Centre of Excellence for Particle Physics at the Tera-scale, School of Chemistry and Physics,
University of Adelaide, Adelaide SA 5005, Australia
http://www.physics.adelaide.edu.au/cssm
2
Department of Physics, School of Science, Tokai University, Hiratsuka-shi, Kanagawa 259-1292, Japan
http://www.sp.u-tokai.ac.jp/
(Dated: August 30, 2012)
The Nambu–Jona-Lasinio jet model is extended to accommodate hadronization of a transversely
polarized quark in order to explore the Collins effect within a multihadron emission framework.
This is accomplished by calculating the polarized quark spin flip probabilities after a pseudoscalar
hadron emission and the elementary Collins functions. The model is then used to calculate the
number densities of the hadrons produced in the polarized quark’s decay chain. The full Collins
fragmentation function is extracted from the sine modulation of the polarized number densities
with respect to the polar angle between the initial quark’s spin and hadron’s transverse momentum.
Two cases are studied here. First, a toy model for elementary Collins function is used to study the
features of the transversely polarized quark-jet model. Second, a full model calculation of transverse
momentum dependent pion and kaon Collins functions is presented. The remarkable feature of our
model is that the 1/2 moments of the favored Collins fragmentation functions are positive and peak
at large values of z but decrease and oscillate at small values of z. The 1/2 moments of the unfavored
Collins functions have comparable magnitude and opposite sign to the favored functions, vanish at
large z and peak at small values of z. This feature is observed for both the toy model and full
calculation and can therefore be attributed to the quark-jet picture of hadronization. Moreover,
the transverse momentum dependencies of the model Collins functions differ significantly from the
Gaussian form widely used in the empirical parametrizations. Finally, a naı̈ve interpretation of
the Schäfer-Teryaev sum rule is proven not to hold in our model, where the transverse momentum
conservation is explicitly enforced. This is attributed to the sizable average transverse momentum
of the remnant quark that needs to be accounted for to satisfy the transverse momentum sum-rule.
PACS numbers: 13.60.Hb, 13.60.Le, 13.87.Fh, 12.39.Ki
Keywords: Collins fragmentation functions, TMDs, NJL-jet model, Monte Carlo simulations
I.
INTRODUCTION
Deep inelastic scattering (DIS) has been a powerful
tool with which to explore the structure of hadrons. Particularly, experiments using the semi-inclusive deep inelastic scattering (SIDIS) have advanced our understanding of the underlying partonic structure of nucleon in momentum space, allowing access to both the longitudinal
and transverse momentum distribution of quarks and gluons. Using polarized probes and/or targets has allowed
us to extend these explorations to the spin dependence
of the parton distributions. In particular, the naively
time-reversal odd Sivers distribution function has been
measured to be nonzero [1–3]. The SIDIS cross sections,
both polarized and unpolarized, are a convolution of parton distribution functions with elementary parton-probe
scattering amplitudes calculable using perturbative QCD
and parton fragmentation functions [4]. The complete
tree level expressions at leading order have been written
in Ref [5]. Thus our detailed understanding of nucleon
structure from SIDIS is hinged on our knowledge of fragmentation functions. Moreover, the naive T-odd distributions are convoluted with T-odd fragmentation functions, notably the Collins fragmentation function H1⊥ .
The first direct measurements of the Collins mechanism
have been performed by the Belle collaboration using
hadron pair production in e+ e− collisions [6, 7].
There has been a vast amount of work done to calculate
the Collins function within theoretical models. Most notably, the first calculation of a nonvanishing Collins function for pion production within a quark spectator model,
using the interference of one-pion-loop amplitudes with
the tree level amplitude, was preformed in Refs. [8, 9].
The resulting Collins fragmentation function appears to
be too small to describe the experimental data as shown
in Ref. [10], where the contributions from initial quark
rescattering were largely canceled by the quark-pion vertex loop corrections. Later, a similar mechanism was
used to calculate the Collins function, through interference of the gluon rescattering and gauge-link correction amplitudes interference with the tree level amplitude
Refs. [10–12]. The resulting Collins functions are in generally good agreement with the data, but this approach
lacks the ability to produce the unfavored fragmentation
functions, including Collins functions. The experimental
results from HERMES, COMPASS and JLab are strongly
suggesting that the unfavored Collins functions have a
similar size and an opposite sign to that of the favored
ones [1, 3, 13, 14].
The NJL-jet model of Refs. [15–18] has been used
to calculate the quark fragmentation function within
the quark-jet hadronization picture of Field and Feyn-
2
man [19], using the elementary quark-hadron splitting
functions calculated within the effective quark model of
Nambu and Jona-Lasinio (NJL) [20]. The advantage of
the model is that it has no free parameters adjusted to
fragmentation data and it naturally describes the unfavored fragmentation functions. The use of Monte Carlo
(MC) methods in Refs. [17, 18] allows for inclusion of
hadronic resonances and their decays as well as the transverse momentum dependence of both parton distribution
and fragmentation functions. The model was also used to
calculate the dihadron fragmentation functions and their
evolution in Refs. [21, 22]. Here we propose to extend the
NJL-jet model by including the spin of the fragmenting
quark to calculate the Collins function, both favored and
unfavored.
Here we consider the fragmentation of the transversely
polarized quark, as depicted schematically in Fig. 1. The
goal is to calculate the T-odd Collins fragmentation function using the NJL-jet Monte Carlo method of Ref. [18].
Following the ”Trento Convention” [23], the fragmentation function of the transversely polarized quark q to unpolarized hadron h carrying the light-cone momentum
fraction z and transverse momentum P⊥ with respect to
quark’s momentum k can be expressed as a sum of two
terms
h/q
Dh/q↑ (z, P⊥2 , ϕ) = D1 (z, P⊥2 )
⊥h/q
+ H1
(z, P⊥2 )
(1)
(k̂ × P⊥ ) · Sq
,
zmh
where k̂ and Sq are the momentum and the spin vector
of the fragmenting quark, mh is the mass of the produced hadron. The unpolarized fragmentation function
h/q
⊥h/q
is denoted as D1 (z, P⊥2 ) and H1
(z, P⊥2 ) is the socalled Collins function. The spin dependent geometric
factor multiplying the Collins function in Eq. (1) can be
expressed in terms of the polar angle ϕ of the vector P
using the 3-vector components, yielding for the polarized
fragmentation function
h/q
Dh/q↑ (z, P⊥2 , ϕ) = D1 (z, P⊥2 )
(2)
P⊥ Sq
⊥h/q
− H1
(z, P⊥2 )
sin(ϕ).
zmh
The measure for the above number density is given as
(3)
dNh/q↑ (z, P⊥ ) = Dh/q↑ (z, P⊥2 , ϕ) dz d2 P⊥
2
P⊥ Sq sin(ϕ)
dP
h/q
⊥h/q
= D1 (z, P⊥2 ) − H1
(z, P⊥2 )
dz ⊥ dϕ.
zmh
2
Integrating over the azimuthal angle ϕ we recover the
expression for the number density of the unpolarized
quark
dNh/q↑ (z, P⊥2 ) ≡
Z
2π
dϕ Nh/q↑ (z, P⊥2 , ϕ)
(4)
0
h/q
= πD1 (z, P⊥2 ) dz dP⊥2 = dNh/q (z, P⊥2 ).
On the other hand, if we perform an integration over
P⊥2 , we arrive at an expression that allows us to extract
⊥(1/2)
the 1/2 moment of the Collins function H1(h/q) (z)
dNh/q↑ (z,ϕ) ≡ Dh/q↑ (z, ϕ) dz dϕ
Z ∞
≡
dP⊥2 Nh/q↑ (z, P⊥2 , ϕ)
(5)
0
i
1 h h/q
⊥(1/2)
=
D1 (z) − 2H1(h/q) (z)Sq sin(ϕ) dz dϕ,
2π
where
Z
h/q
∞
h/q
dP⊥2 D1 (z, P⊥2 ),
D1 (z) ≡ π
(6)
0
Z
⊥(1/2)
H1(h/q) (z) ≡ π
∞
dP⊥2
0
P⊥
⊥h/q
H
(z, P⊥2 ).
2zmh 1
(7)
At last, integrating over both P⊥2 and ϕ in Eq. (3) we
arrive at the integrated number density
Z
dNh/q↑ (z) ≡
0
∞
dP⊥2
Z
2π
dϕ Nh/q↑ (z, P⊥2 , ϕ)
(8)
0
h/q
= D1 (z) dz.
FIG. 1. Illustration of the three dimensional kinematics of
transversely polarized quark fragmentation. The fragmenting
quark’s momentum k defines the z-axis with its transverse
polarization spin vector Sq along x axis. The emitted hadron
has momentum P with the transverse component P⊥ with
respect to the z-axis. The polar angle of hadron’s momentum
P with respect to the zx plane is denoted by ϕ.
Our strategy here is to use the NJL-jet model to calculate the number densities Nh/q↑ (z, P⊥2 , ϕ) and use the dependence on the azimuthal angle ϕ in Eq. (3) to extract
h/q
⊥h/q
both D1 (z, P⊥2 ) and H1
(z, P⊥2 ), where the former
can be compared with our earlier results in Ref. [18] as
a cross-check. To achieve this goal, we first expand the
NJL-jet model and the associated Monte Carlo framework to accommodate the transverse spin of the fragmenting quark in Sec. II. In Sec. III, we calculate the
3
elementary unpolarized and Collins fragmentation functions, needed as an input to the NJL-jet model. In
Sec. IV, we present the results for a simple toy model
used as elementary Collins function to demonstrate the
distinctive features of the model. In Sec. V, we discuss
the full model results and present the conclusions and
some final remarks in Sec. VI.
II.
NJL-JET WITH TRANSVERSELY
POLARIZED QUARK
The NJL-jet model employs the quark-jet mechanism
to describe the hadronization process. Here we use the
model to describe the fragmentation of the transversely
polarized quarks to unpolarized hadrons, schematically
depicted in Fig. 2.
An important intricacy arises from the need to keep
track of the transverse spin of the quark in the jet as
it emits hadrons. We first examine the elementary process where a transversely polarized quark emits a single hadron, and we calculate the probability of the final
quark’s spin being parallel or antiparallel to the original quark’s spin direction. In this work we only consider
the emissions of pseudoscalar mesons. Further, we make
a first order approximation by including in the calculations of the quark spin flip probabilities the elementary
hadron emissions only via the tree level diagram depicted
in Fig. 4, thus neglecting any T-odd effects. The T-odd
effects are considered here to be small relative to the
included unpolarized term, though essential for generating the elementary Collins function. This approximation
can be easily improved on in the future by also including
in the quark spin flip calculation the relevant diagrams
that generate the elementary Collins function, such as
that discussed in Sec. III B.
We use the kinematics depicted in Fig. 1, with the
fragmenting quark’s momentum defining the z axis. We
denote the remnant quark as Q with momentum and spin
vectors l and SQ . We use the Dirac spinors of [24, 25]
to describe the wave functions of transversely polarized
quarks
1 Uχ ≡ √ U(+z) + χ U(−z) ,
2
where χ = ±1 are the eigenvalues of the spin projection onto the x axis and U(±z) are the Lepage-Brodsky
spinors in helicity basis [26]. Note that we have a plus
sign in front of the second term with χ, as there is a
sign error in the corresponding expressions of [24]. The
spinors Uχ are both solutions of the Dirac equation and
the eigenstates of the Lorentz-covariant spin operator:
the Pauli-Lubanski vector
1
Wµ ≡ − µνρσ S νρ k σ ,
2
ı
S νρ ≡ [γ ν , γ ρ ] ,
4
(10)
(11)
where µνρσ is the Levi-Civita tensor with convention
0123 = +1. It is easy to check explicitly, that
m
Uχ ,
2
(/
k − m) Uχ = 0.
W1 U χ = χ
(12)
(13)
The normalization of the spinors is
Ūχ (k, m)Uχ0 (k, m) = δχ,χ0 2m.
(14)
Then the fragmenting quark’s spinor is Ψin = U1 (k, M1 ),
while the remnant quark’s spinor can be described as
a superposition of states with spin polarization parallel
and antiparallel to the x axis: Ψout = a1 U1 (l, M2 ) +
a−1 U−1 (l, M2 ). Then the relative probabilities of
the
quark spin flip and nonflip are determined by
Ψ̄out γ 5 Ψin 2 . The spinor matrix elements are given by
Ūχ0 (l, M2 )γ 5 Uχ (k, M1 )2 =
δχ,χ0
lx2
1−z
+ δχ,−χ0
(15)
ly2
+ (M2 − (1 − z)M1 )2
,
1−z
and hence the spin nonflip and spin flip probabilities are
proportional to
|a1 |2 ∼ lx2 , |a−1 |2 ∼ ly2 + (M2 − (1 − z)M1 )2 .
FIG. 2. NJL-jet model including transverse momentum and
quark polarization transfer. Here the orange double-lined arrows schematically indicate the spin direction of the quark in
the decay chain.
(9)
(16)
We note, however, that the spinors Uχ (k, m) become
the eigenstates of W1 only if the momentum k has no
transverse components: kx = ky = 0. Though the final state quark necessarily has a transverse momentum
l⊥ ≡ (lx , ly ) = −p⊥ , required by the momentum conservation in the transverse plane, it is assumed to be
small compared to the light-cone momentum. Thus, in
each hadron emission step, we assume the fragmenting
quark being in an eigenstate Uχ , and sample the spin
flip probabilities according to (16). It will be discussed
in Secs. IV and V that the results for the Collins function simulations converge rapidly with the number of the
hadron emissions in the region of z & 0.02, similar to the
4
FIG. 3. Quark elementary fragmentation kinematics, for an
arbitrary hadron emission in the cascade chain. The z axis
is defined by the direction of the 3-momentum of the original
parent quark.
case of the unpolarized fragmentation function that was
examined in Ref. [17]. Thus the failure of this approximation after many hadron emissions, when the light-cone
momentum of the quark becomes small and comparable
to the transverse component, affects the results only at
extremely small values of z and can be neglected.
Using the probabilities of the quark spin flip of Eq. (16)
and the elementary polarized number densities, to be calculated in the next section, we perform a Monte Carlo
simulation within the NJL-jet framework to calculate
the average multiplicities of the produced pseudoscalar
mesons (pions and kaons), Nh/q↑ , as functions of light-
Dh/q↑ (z, P⊥2 , ϕ) ∆z
cone momentum fraction z, transverse momentum square
P⊥2 , and polar angle ϕ, in the coordinate system defined
by the initial quark momentum and spin polarization.
Schematically, the elementary hadron emission process is
depicted in Fig. 3, where the vectors k and k0 denote the
3-momentum of an arbitrary quark in the cascade chain
before and after hadron emission with transverse com0
ponents k⊥ and k⊥
, respectively. The emitted hadron’s
momentum is labeled by ph , where its transverse component with respect to k and the z axis is denoted by p⊥
and P⊥ , respectively. P⊥ is obtained using the relation
P⊥ = p⊥ + z k⊥ . The recoil transverse momentum of
0
the final quark, k⊥
, is calculated from momentum conservation in the transverse plane, namely,
0
k ⊥ = P⊥ + k ⊥
.
(17)
In each hadron emission step we randomly sample the
type, the relative light-cone momentum fraction z and
transverse momentum and polar angle ϕ (relative to the
spin vector of the fragmenting quark) for the produced
hadron according to the relevant elementary number density. We recover and record these quantities in the initial
quark’s frame, as well as sample the spin direction of the
final quark according to (16). Then we can extract the
number densities corresponding to Eq. (3) in the usual
way [17, 18]
D
E
∆P⊥2
∆ϕ = Nqh↑ (z, z + ∆z; P⊥2 , P⊥2 + ∆P 2 ; ϕ, ϕ + ∆ϕ)
2
P
h
2
2
2
NSims Nq ↑ (z, z + ∆z, P⊥ , P⊥ + ∆P⊥ ; ϕ, ϕ + ∆ϕ)
≡
,
NSims
(18)
where Nqh↑ is the number of the hadrons h produced by the quark q that have momentum components laying within
the regions specified in its arguments and NSims is the number of quark decay chain simulations performed.
III.
ELEMENTARY FRAGMENTATIONS
The quark fragmentation functions are defined using the quark-quark correlators [4, 11, 27]
X Z dξ − d2 ξT
eık·ξ 0| U T [∞T , ξT ; −∞− ] U − [−∞− , ξ − ; ξT ] ψ(ξ)|h, X
∆(z, P⊥ ) =
4z(2π)3
X
× h, X| ψ̄(0) U − [0− , −∞− ; 0T ] U T [0T , ∞T ; −∞− ]|0 ξ+ =0 ,
where ψ is the quark wave function, and U (−,T ) represents the gauge links along the minus and transverse
directions in light-cone coordinates. The details of the
gauge link structure can be found in Ref. [27].
In this section we aim to calculate the elementary unpolarized and Collins fragmentation functions (or splitting functions), corresponding to only a single hadron
(19)
emission truncation of the relation (19). We use the
framework of the NJL effective chiral quark model for
this purpose. The corresponding unpolarized splitting
functions, have been calculated in Refs. [15–17], where
the model parameters were also fixed. In the next subsection we briefly summarize their results and refer the
reader to the original articles for the details. For the
5
p
2
where M12
is their invariant mass squared. In the frame
2
where the total transverse momentum is zero, M12
=
2
2
2
2
(µ1 + p⊥ )/x + (µ2 + p⊥ )/(1 − x). The cutoff, Λ12 , is
determined by
q
2
q
2
M12
≤ Λ212 ≡
Λ23 + µ21 + Λ23 + µ22 ,
(23)
p
k
k
kïp
FIG. 4. Feynman diagram describing the elementary quark
to hadron fragmentation functions.
Collins splitting functions, we adopt the gauge link coupling mechanism to the final quark of Refs. [10–12], but
use the NJL model formalism and parameters.
A.
Unpolarized splitting function
where Λ3 is the 3-momentum cutoff, that is fixed in
the usual way by reproducing the experimental pion decay constant. We use a light constituent quark mass
M = 0.4 GeV, for which the corresponding 3-momentum
cutoff is Λ3 = 0.77 GeV. The strange constituent quark
mass of Ms = 0.59 GeV is obtained by reproducing
the experimental kaon mass. The model values for the
quark-meson coupling constants are determined from the
residue at the pole in the quark-antiquark t matrix to be
gπqQ = 4.24 and gKqQ = 4.52.
h/q
The unpolarized fragmentation function, D1 (z, P⊥ ),
is defined as a trace of the quark-quark correlator
D1 (z, P⊥2 ) = Tr[∆[z, P⊥ ]γ + ]/2.
(20)
The single hadron emission cut diagram for a quark
q, to emit a meson h, carrying light-cone momentum
fraction z, and transverse momentum p⊥ , is depicted in
Fig. 4. In the frame where the fragmenting quark has zero
transverse momentum, but a nonzero transverse momentum component −p⊥ /z with respect to the direction of
the produced hadron [15, 28], the unregularized elementary transverse-momentum-dependent (TMD) fragmentation functions to pseudoscalar mesons are given by
h/q
d1 (z, p2⊥ ) =
×
Cqh 2
g
z
16π 3 hqQ
2
p2⊥ + [(z − 1)M1 + M2 ]
(21)
2,
[p2⊥ + z(z − 1)M12 + zM22 + (1 − z)m2h ]
where M1 and M2 denote the masses of fragmenting and
remnant quarks. Quark flavor is also indicated by the
subscripts q and Q, where a meson of type h has the
quark flavor structure h = qQ and mh denotes the meson mass. The corresponding isospin factor and quarkmeson coupling constant are labeled by Cqh and ghqQ ,
respectively, and have been determined within the NJL
model [16, 18].
In the model, the transverse momentum integrated
fragmentation of Eqs. (6,7) (as well as the quark-hadron
couplings, etc.) require an introduction of a regularization scheme. In this work, the integrals with two particles
in the loop with masses µ1 and µ2 , carrying light-cone
momentum fractions x and 1 − x, respectively, are regularized using the modified Lepage-Brodsky transverse
momentum cutoff functions, as described in [17]
G12 (p2⊥ ) ≡
1
2,
2 /Λ2 )2 ]
[1 + (M12
12
(22)
B.
Elementary Collins function
The Collins function is defined via the trace of the
quark-quark correlator
ij
T kT j
H1⊥ (z, P⊥2 ) = Tr[∆(z, P⊥ )ıσ i− γ5 ],
mh
(24)
where mh is the mass of the produced hadron, 12
T =
11
22
−21
=
1,
and
=
=
0
.
T
T
T
It was shown explicitly, that the simple tree level approximation for the correlator, used to calculate the unpolarized quark splitting function, yields a zero result for
the Collins function [8, 10]. Thus, here we generate the
model Collins function through the interference between
the tree level hadron emission amplitude and the one with
a gauge-link interaction with the final state quark via single gluon exchange, as described in Ref. [11]. Note, that
we omit the gluon-quark rescattering and gluon loop correction to the quark-hadron vertex, as those nearly cancel each other. Also, we note that the amplitudes written
in the spectator formalism coincide with those in NJL,
except for the different quark-hadron coupling constant,
the remnant ”spectator” being treated as a quark in NJL
and the loop integral regularization scheme employed in
the NJL-jet. Thus we can simply use the results from
Ref. [11] with just a few modifications. The relevant diagram for the NJL-jet model is depicted in Fig. 5.
The result for the elementary Collins function is
Cm
mh
1
z
e ⊥ (z, p2 ) = − 2αs CF q g 2
H
1
⊥
(2π)4
2 mqQ 1 − z k 2 − M12 2p2⊥
× {−I34g [(2z − 1)M1 + M2 ]
+ I2g 2zM1 (k 2 − M12 + m2h (1 − 2/z))
+ (M2 − M1 )((2z − 1)k 2 − m2h
+M22 − 2zM1 (M1 + M2 )) },
(25)
where αs is the strong coupling constant, CF = 4/3 is
the cubic Casimir invariant of the color SU (3) group and
6
p
1.2
p
u π+
u K+
s K-
1.0
k
k−l
0.8
k−p
d1
l
0.6
0.4
FIG. 5. Cut diagram describing the elementary quark to
hadron Collins fragmentation functions.
0.2
0
0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
z
2
the initial quark’s total momentum squared, k , can be
expressed as k 2 = p2⊥ /(z(1 − z)) + M22 /(1 − z) + m2h /z.
Since only the imaginary part of the amplitudes contribute to the above expression, the loop integrals for
those can be calculated without the need for explicit regularization
#
"
p
k 2 + M22 − m2h − λ(M2 , mh )
π
p
I2g = p
,
ln
2 λ(M2 , mh )
k 2 + M22 − m2h + λ(M2 , mh )
"√
#
k 2 (1 − z)
I34g = π ln
,
(26)
M2
(a)
0.6
u π+
u K+
s K-
0.5
2 h1⊥ (1/2)
0.4
0.3
0.2
0.1
λ(m1 , m2 ) = (k 2 − (m1 + m2 )2 )(k 2 − (m1 − m2 )2 ).
0
0
0.2
0.4
z
Then the elementary polarized fragmentation function
is given as
(b)
h/q
e ⊥h/q (z, p2⊥ ) p⊥ Sq sin(ϕ),
dh/q↑ (z, p2⊥ , ϕ) = d1 (z, p2⊥ ) − H
1
zmh
(27)
FIG. 6. Elementary renormalized unpolarized fragmentation
⊥(1/2)
,
function, dˆ1 , (a) and Collins function 1/2 moment, 2Ĥ1
(b) used in the full model calculations of Sec. V.
where we use the multiplicative regulator of Eq. (22) for
integrals over p2⊥ . The only remaining parameter is the
strong coupling in Eq. (25). We consider it as a model
parameter, that we fix to the largest value of αs = 0.444,
that still allows for the positivity bound to be satisfied,
namely dh/q↑ > 0. At the next-to-leading order, this
value corresponds to a typical hadronic scale of Q2 =
1 GeV2 , which is much higher than the one typically
used as the NJL-jet model scale, namely Q20 = 0.2 GeV2 .
Such a discrepancy between the scales for the unpolarized and Collins functions is rooted in the model for the
Collins function employed here, which would violate the
positivity bound if calculated at the typical model scale.
A similar issue was encountered in the original work of
Ref. [11], where the value of αs = 0.2 was chosen, much
smaller than that for the scale of their model set at
Q2 = 0.4 GeV2 . A completely consistent determination
of a single scale for the polarized fragmentation function
must involve the QCD evolution of both the TMD unpolarized and Collins functions. This is not possible at
present, because the evolution equation for the Collins
function is unknown.
Within the NJL-jet model, the elementary splitting functions are renormalized such that quark’s total
probability of emitting a hadron in each step is one:
dz dp2⊥ /2 dϕ dˆh/q↑ (z, p2⊥ , ϕ) = 1, the sum is over
all hadrons the quark of given flavor can emit directly.
These renormalized splittings will be used in the next two
sections as input to the Monte Carlo simulations of the
quark-jet hadronization process. The integrated renormalized elementary fragmentation functions for the full
model calculations of Sec. V are depicted in Fig. 6.
P R
h
IV.
THE QUARK-JET EFFECTS ON COLLINS
FUNCTION USING A TOY MODEL
In this section we employ a toy model for the elementary Collins function to explore the general features of the
NJL-jet model extended to transversely polarized quark
fragmentation, as described in Sec. II. In this toy model
e ⊥h/q (z, p2 ) p⊥ Sq = 0.1 dh/q (z, p2 ).
we assume that −H
1
1
⊥ mh z
⊥
Thus for the elementary number density we simply have
(toy)
h/q
dh/q↑ (z, p2⊥ ) = d1 (z, p2⊥ )(1 + 0.1 sin ϕ),
h/q
(28)
where d1 (z, p2⊥ ) is given by Eq. (21).
For the toy
model, we perform the MC simulations with only light
7
z=0.2
u h
0.8
0.908
π0
π+
π-
0.906
0.6
z D1
Dh/q↑
0.904
0.902
0.4
0.9
0.898
0
0.2
1
2
3
ϕ
4
5
6
0
z=0.6
0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.6
0.8
1.0
z
0.36
(a)
Dh/q↑
0.35
u h
0.34
π0
π+
π-
0.10
0.33
0.05
0.31
0
2 H1⊥ (1/2)
0.32
1
2
3
ϕ
4
5
6
0
FIG. 7. The histograms (blue dots) for polarized number
density Dπ0 /u↑ (z, ϕ) for two values of z as a function of the
azimuthal angle ϕ from NJL-jet framework using a toy model
with a fixed number of hadrons emitted in each decay chain
NLinks = 6. The minimum-χ2 fits with a functional form
Dh/q↑ = c0 + c1 sin(ϕ) (red lines) are also depicted.
−0.05
0
0.2
0.4
z
(b)
u h
π0
π+
π-
0.10
2 H1⊥ (1/2) / D1
quarks in the quark-jet and pions in the final state, for
simplicity. We perform several high-statistics simulations, where the number of produced hadrons in each
decay chain, NLinks , is fixed to a particular value. We
use the relation (18) to extract the polarized number
density Dh/q↑ (z, P⊥2 , ϕ) from the numbers of produced
hadrons in intervals of the variables z, P⊥2 and ϕ. For
simplicity, in this section we will only consider the results for the P⊥2 -integrated polarized quark fragmentation
Dπ+ /u↑ (z, ϕ) that can be obtained from Dh/q↑ (z, P⊥2 , ϕ)
using the relation in Eq. (5). Next, to extract the unpolarized and the 1/2 moment of the Collins functions
⊥(1/2)
D1 (z) and H1
(z), we perform a minimum-χ2 fit to
Dh/q↑ (z, ϕ) using a form F (c0 , c1 ) ≡ c0 + c1 sin(ϕ) for
fixed values of z. This fitting method allows us to better account for the statistical fluctuations in the MC results when extracting the Collins function. Notably, the
fits describe the produced functions very well, yielding
χ2 over the number of degrees of freedom always in the
vicinity of 1. A sample of such extraction for u → π 0
fragmentation is shown in Fig. 7, for simulations with
NLinks = 6. The histograms (blue dots) show the polarized number density Dπ+ /u↑ (z, ϕ) for two values of z as a
function of the azimuthal angle ϕ. The minimum-χ2 fits
0.05
0
−0.05
0
0.2
0.4
z
(c)
FIG. 8. Fitted values for unpolarized fragmentation function,
⊥(1/2)
D1 , (a) and twice the Collins function 1/2 moment, 2H1
,
0
+
−
(b) and their ratio (c) for π , π and π versus z, produced
by u quark from quark-jet framework using a toy model. The
error bars show the uncertainties from the statistics and fits,
and are only visible for the unfavored ratio at large values of
z.
8
with a functional form Dh/q↑ = c0 + c1 sin(ϕ) (red lines)
are also depicted.
⊥(1/2)
Then, using D1 (z) = 2πc0 and −2Sq H1
(z) =
2πc1 [see Eq. (5)], we plot the resulting fragmentation
functions for u quark and NLinks = 6 as function of z
in Fig. 8, setting, for example, Sq = −1. The plots in
Fig. 8(a) depict the extracted, integrated fragmentation
functions, which are in agreement with the results of the
direct MC calculations in our previous work [16–18].
The results for the moment of the Collins function,
⊥(1/2)
2H1
, depicted in Fig. 8(b), exhibit very interesting
features: both functions for u → π + and u → π 0 , though
peak at values of z ∼ 0.6, decrease and u → π 0 changes
the sign for small values of z. The unfavored function
u → π − , generated solely by multihadron emission, has
opposite sign and peak value at small z of comparable
size to that of the favored ones at large z. A similar
⊥(1/2)
picture is found for the ratio 2H1
/D1 , depicted in
0.10
2 H1⊥ (1/2)
0.05
0
u h
π +, NL=1
π +, NL=2
π -, NL=2
π +, NL=6
π -, NL=6
−0.05
−0.10
0
0.2
0.4
0.6
0.8
1.0
z
(a)
u h
2 H1⊥ (1/2) / D1
V.
COLLINS FUNCTION FOR PIONS AND
KAONS
In this section we present the results of the full MC
simulations with light and strange fragmenting quarks as
well as pions and kaons as the produced hadrons. The elementary splitting functions were taken from Eq. (27) and
the simulations were done for values of NLinks equal to 1,
2 and 6. Again, it was checked with lower statistics runs,
that the solutions are indistinguishable for NLinks > 6
with the number of discretization points for z, P⊥2 and ϕ
used here.
0.10
π +, NL=1
π +, NL=2
π -, NL=2
π +, NL=6
π -, NL=6
0.05
Fig. 8(c), where notably the results for π 0 are below those
for π + , while these ratios for the corresponding elementary functions coincide.
To better understand the results for the u → π + and
u → π − Collins functions, in Fig. 9 we depict the results
⊥(1/2)
for NLinks equal to 1, 2 and 6. The plots for 2H1
in
Fig. 9(a) show that the functions change very little for
NLinks > 2 in the high z region. That is, the bulk of the
model effects for z & 0.2 can be described with just two
hadron emissions. We verified numerically with lower
statistics runs, that the further increase in NLinks > 6
affects the functions only at extremely low values of z, below the numerical discretization size of ∆z = 0.01 used
in this study. A naı̈ve interpretation for the results is
that the remnant quark has larger probability to have
its spin antiparallel to that of the splitting quark, see
Eq. (16), and typically has a small fraction of the initial
light-cone momentum. Then in the emission step of the
second hadron that affects the low z region and generates the unfavored function, the angle ϕ in the the Collins
term in Eq. (27) in most cases will acquire an additional
π phase, yielding the results in Fig. 9. This picture repeats for further hadron emissions, creating ”destructive
interference” in the small-z region, decreasing the Collins
⊥(1/2)
functions. The ratio 2H1
/D1 , depicted in Fig. 9(b),
also exhibits the contrast in the low-z behavior of the
unpolarized and Collins functions: while D1 (z) grows
roughly as 1/z, the Collins function oscillates to 0. It is
worth mentioning that, for NLinks = 1, the fitted value
of the ratio in u → π + equals to the one set in the toy
model as input to the MC.
0
A.
−0.05
0
0.2
0.4
0.6
0.8
1.0
z
(b)
⊥(1/2)
FIG. 9.
Fitted values for 2H1
(a) and the ratio
⊥(1/2)
2H1
/D1 (b) for π + and π − versus z, produced by u
quark from quark-jet framework using a toy model for increasing values of NL ≡ NLinks .
P⊥2 integrated results
We first present the results for the P⊥2 -integrated
Collins function, skipping the results for the unpolarized
functions, D1 (z), as they have been studied in detail in
our previous work.
⊥(1/2)
⊥(1/2)
The results for 2H1
and the ratio 2H1
/D1 for
the hadrons produced by a u quark are shown in Figs. 10
⊥(1/2)
and 11, respectively. The fitted values for 2H1
for
pions show similar features to those from the toy model:
the favored functions, positive and peaking at z ∼ 0.65,
9
u h
0.3
u h
π0
π+
π-
2 H1⊥ (1/2) / D1
0.2
2 H1⊥ (1/2)
π0
π+
π-
0.3
0.1
0.2
0.1
0
0
−0.1
−0.1
0
0.2
0.4
0.6
0.8
1.0
0
0.2
0.4
z
(a)
0.3
K0
K+
K-
1.0
0.6
0.8
1.0
u h
K0
K+
K-
0.2
2 H1⊥ (1/2) / D1
0.05
2 H1⊥ (1/2)
0.8
(a)
u h
0.10
0.6
z
0
0.1
0
−0.1
−0.2
−0.05
0
0.2
0.4
0.6
0.8
1.0
−0.3
0
0.2
0.4
z
z
(b)
⊥(1/2)
(b)
⊥(1/2)
for pions (a) and kaons
FIG. 10. Fitted values for 2H1
(b), produced by u quark for NLinks = 6.
/D1 , for pions
FIG. 11. The ratio of the fitted values, 2H1
(a) and kaons (b), produced by u quark for NLinks = 6.
decrease and some change sign at lower values of z, oscillating around zero as z → 0. The unfavored function,
u → π − , is mostly negative and peaking at z ∼ 0.2, with
a peak value about one-third of that for u → π + . The results for kaons are similar to the those for pions, with the
exception of u → K + . This is a favored fragmentation,
but the peak value of the unfavored function u → K − is
only slightly less of that for the favored fragmentation.
Also, since in our model charge and isospin symmetries
are exact, the results for u → K̄ 0 coincide with the ones
for u → K − and are omitted.
Similar results for the s quark are shown in Fig. 12.
⊥(1/2)
for pions and unfavored
The fitted values for 2H1
kaon channels peak at value of z ' 0.1 with values much
smaller than those for the favored channels of K − and
K̄ 0 , which have a very broad peak and do not become
negative at any value of z.
The dependence on NLinks can be seen in Fig. 13.
Again, as for the toy model, we notice that the results
change very little for z > 0.2 for NLinks > 2. The
increase in NLinks affects the functions at decreasingly
smaller values of z.
B.
TMD of Collins function
In this section we present the full results for the TMD
Collins function. As the z dependence has been studied in detail in the previous sections, here we will only
show the P⊥2 dependence of the Collins function for several fixed values of z. We noticed that the polar angle
dependencies of the polarized number densities are not
very accurately described by just the sine modulation
term corresponding to the Collins term for some values
of z and P⊥2 . In our calculations with NLinks = 6, we
found that a fourth order polynomial in sin(ϕ)
Dh/q↑ (z, P⊥2 , ϕ) =
4
X
cn (z, P⊥2 ) sinn ϕ
(29)
n=0
is sufficient to achieve fits for all slices with fixed z, P⊥2
with χ2 per degree of freedom in the vicinity of 1. Here we
10
s h
0.6
u h
π+
K+
K-
0.5
6
π 0, z=0.7
π +, z=0.7
π -, z=0.7
4
0.3
D1
2 H1⊥ (1/2)
0.4
0.2
2
0.1
0
0
−0.1
0
0.2
0.4
0.6
0.8
1.0
0
0.2
(a)
0.8
1.0
14
u h
π+
K+
K-
0.5
0.6
(a)
s h
0.6
0.4
P⊥2 (GeV 2)
z
12
π 0, z=0.1
π +, z=0.1
π -, z=0.1
10
8
0.3
D1
2 H1⊥ (1/2) / D1
0.4
0.2
0.1
4
0
2
0
−0.1
0
0.2
0.4
0.6
0.8
1.0
(b)
⊥(1/2)
⊥(1/2)
/D1 (b),
u h
K +, NL=1
K +, NL=2
K -, NL=2
K +, NL=6
K -, NL=6
0.05
0
−0.05
0
0.2
0.2
0.4
0.6
0.8
1.0
(b)
(a) and 2H1
FIG. 12. Fitted values for 2H1
for hadrons produced by s quark for NLinks = 6.
0.10
0
P⊥2 (GeV 2)
z
2 H1⊥ (1/2)
6
0.4
0.6
0.8
1.0
z
⊥(1/2)
FIG. 13. Fitted values for 2H1
for K + and K − versus
z, produced by u for increasing values of NL ≡ NLinks .
FIG. 14. The unpolarized fragmentation function vs P⊥2 for
pions produced by a u quark with z = 0.7 (a) and z = 0.1 (b)
for NLinks = 6.
identify c0 with the unpolarized and c1 with the Collins
term of Eq. (2). The higher powers of the sin(ϕ) in the
polarized quark fragmentation functions are induced by
the multiple hadron emissions. A toy model analysis
showed that there are two sources for these effects: the
remnant quark transverse momentum distribution modulation from the recoil in the elementary hadron emissions,
and an additional modulation of the same quark distribution due to the ϕ dependence of the quark spin flip
probabilities. The detailed description and explanation
of these additional terms are presented in our forthcoming publications [29, 30], where we show that these effects
are a genuine feature of the quark-jet model and do not
depend on the particular form of the elementary polarized fragmentation function. The plots in Figs. 14, 15
and 16 depict the results for unpolarized, Collins fragmentation functions and their ratios respectively for pions produced by an initial u quark with two fixed values
of z equal to 0.7 and 0.1. For z = 0.7, the results are
mostly affected by a single hadron emission, thus the P⊥2
11
u h
0.4
π 0, z=0.7
π +, z=0.7
π -, z=0.7
u h
π 0, z=0.7
π +, z=0.7
π -, z=0.7
P⊥
H ⊥ / D1
z mh 1
1.0
P⊥
H⊥
z mh 1
0.2
0.5
0
0
0
0.2
0.4
P⊥2
0.6
0.8
1.0
−0.2
0
0.2
2
0.4
0.6
0.8
1.0
P⊥2 (GeV 2)
(GeV )
(a)
(a)
0.10
u h
u h
π 0, z=0.1
π +, z=0.1
π -, z=0.1
π 0, z=0.1
π +, z=0.1
π -, z=0.1
0.05
P⊥
H⊥
z mh 1
P⊥
H ⊥ / D1
z mh 1
0.5
0
0
−0.05
0
0.2
0.4
P⊥2
0.6
0.8
1.0
−0.10
0
2
0.2
0.4
0.6
0.8
1.0
P⊥2 (GeV 2)
(GeV )
(b)
(b)
FIG. 15. The Collins fragmentation function vs P⊥2 for pions
produced by a u quark with z = 0.7 (a) and z = 0.1 (b) for
NLinks = 6.
FIG. 16. The ratio of Collins fragmentation function to unpolarized fragmentation function vs P⊥2 for pions produced by
a u quark with z = 0.7 (a) and z = 0.1 (b) for NLinks = 6.
dependence of the Collins function is somewhat similar
to one of the unpolarized fragmentations (which, in turn
are reasonably well described via Gaussian function for
small values of P⊥2 at a fixed z, see [18]), but peak at
a small nonzero value of P⊥2 . The results are quite different for z = 0.1, where the unpolarized functions for
all the pions are roughly equal, while the Collins function for all the pions have similar shapes but differ in
magnitude, decreasing to a negative value and peaking
at P⊥2 ∼ 0.1GeV2 , then start to increase and become
positive for small values of P⊥2 . Thus, it follows from
our model that while the transverse momentum dependence of the unpolarized fragmentation functions can be
to a good approximation described by a Gaussian function with z-dependent width, the Collins functions have
much different shapes.
The plots in Figs. 17, 18 and 19 depict the analogous
results, but for kaons produced by an initial u quark.
The results are similar to those for the pions, except for
the magnitudes of the unpolarized and Collins functions
being smaller.
The plots in Figs. 20, 21 and 22 depict the results for
unpolarized, Collins fragmentation functions and their
ratios respectively for pions and kaons produced by an
initial s quark with two fixed values of z equal to 0.7 and
0.1. In our model the isospin symmetry is considered to
be exact, thus the calculated values for all the pions are
equal to each other, thus only π + is depicted. Similarly,
the results for the kaons that can be transformed to each
other by an isospin rotation coincide and are omitted as
well. The results are similar to those for the u quark,
except for the strong dominance of the favored fragmentation functions at z = 0.7. At z = 0.1, where multiple
hadron emission contributions dominate, the unfavored
fragmentation functions become larger than the favored
ones, in particular for the production of pions.
12
u h
1.5
u h
K 0, z=0.7
K +, z=0.7
K -, z=0.7
K 0, z=0.7
K +, z=0.7
K -, z=0.7
0.2
D1
P⊥
H⊥
z mh 1
1.0
0.1
0.5
0
0
0
0.2
0.4
0.6
0.8
1.0
0
0.2
0.4
0.6
P⊥2 (GeV 2)
P⊥2 (GeV 2)
(a)
(a)
2.5
1.0
0.15
u h
u h
K 0, z=0.1
K +, z=0.1
K -, z=0.1
2.0
K 0, z=0.1
K +, z=0.1
K -, z=0.1
0.10
0.05
D1
P⊥
H⊥
z mh 1
1.5
1.0
0.5
0
0.8
0
−0.05
0
0.2
0.4
P⊥2
0.6
0.8
−0.10
1.0
0
0.2
0.4
2
0.6
0.8
1.0
P⊥2 (GeV 2)
(GeV )
(b)
(b)
FIG. 17. The unpolarized fragmentation function vs P⊥2 for
kaons produced by a u quark with z = 0.7 (a) and z = 0.1
(b) for NLinks = 6.
FIG. 18. The Collins fragmentation function vs P⊥2 for kaons
produced by a u quark with z = 0.7 (a) and z = 0.1 (b) for
NLinks = 6.
C.
The Schäfer-Teryaev sum rule
The Schäfer-Teryaev sum rule was originally proposed
in Ref. [31], where the authors used the simple arguments of the transverse momentum conservation in the
hadronization process to derive
STq ≡
XZ
h
0
1
⊥(1)
dz H1,(h/q) (z) = 0,
(30)
where the first moment of the Collins function is defined
as
Z ∞
P⊥2
⊥(1)
⊥h/q
H1,(h/q) (z) ≡ π
dP⊥2
(z, P⊥2 ).
(31)
H
2zmh 1
0
Later, in Ref. [32], it was proven explicitly using
the quark correlator functions, along with the quark
to hadron Collins function moments that the quark-toquark Collins functions should also be included for the
sum rule to be satisfied
XZ
XZ
⊥(1)
⊥(1)
dz H1,(h/q) (z) +
dz 2H1,(Q/q) (z) = 0. (32)
h
Q
Our model results both for the toy model and the full
calculation, show that the naive Schäfer-Teryaev sum
rule cannot be satisfied, even though the transverse momentum conservation is explicitly satisfied in our simulations. For example, the sum of the terms in Eq. (30)
for u quark is STu = 0.07, while for s quark STs = 0.21.
A simple explanation for this is that with multiple
hadron emissions the quark quickly loses almost all of its
initial light-cone momentum, but gains a nonzero average transverse momentum from the recoil of the emitted
hadrons. This can be easily seen from the fact that the
solutions of unpolarized integrated fragmentation functions, D1 (z), change only in the extremely small region
of z, after only a few hadron emissions, as described in
Ref. [17]. On the other hand, the average transverse momentum of the emitted hadrons tends to level off at a
13
0.4
s h
0.3
K 0, z=0.7
K +, z=0.7
K -, z=0.7
0.1
D1
P⊥
H ⊥ / D1
z mh 1
4
u h
0.2
π +, z=0.7
K +, z=0.7
K -, z=0.7
0
2
−0.1
−0.2
0
−0.3
0
0.2
0.4
0.6
0.8
1.0
0
0.4
0.6
P⊥2 (GeV 2)
(a)
(a)
0.1
0.8
1.0
14
u h
s h
12
K 0, z=0.1
K +, z=0.1
K -, z=0.1
π +, z=0.1
K +, z=0.1
K -, z=0.1
10
8
D1
P⊥
H ⊥ / D1
z mh 1
0.2
P⊥2 (GeV 2)
0
6
4
2
0
−0.1
0
0.2
0.4
0.6
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
P⊥2 (GeV 2)
P⊥2 (GeV 2)
(b)
(b)
FIG. 19. The ratio of Collins fragmentation function to unpolarized fragmentation function vs P⊥2 for kaons produced
by a u quark with z = 0.7 (a) and z = 0.1 (b) for NLinks = 6.
FIG. 20. The unpolarized fragmentation function vs P⊥2 for
pions and kaons produced by a s quark with z = 0.7 (a) and
z = 0.1 (b) for NLinks = 6.
certain nonzero value as z → 0, as shown in Fig. 14 of
[18], hinting at a similar value for the average transverse
momentum of the remnant quark. In the NJL-jet model
we assume that the remnant quark hadronizes with the
slow, colored fragments of the initial struck hadron in
SIDIS, or the remnant of the antiquark in e+ e− reactions, with the product unobserved because of the small
values of remaining z it carries. Thus, while the effects of
the remnant quark after multiple hadron emissions can
be neglected when considering the longitudinal momentum sum rules for the unpolarized fragmentations, they
are essential for the transverse momentum sum rule for
the Collins function, as shown in [32].
VI.
CONCLUSIONS
In this article we calculated the Collins fragmentation
function in the NJL-jet model. This was accomplished
by extending the model to include the transverse polarization of the fragmenting quarks and calculating the
probability of the quark spin flip at each hadron emission. Then, we extended our Monte Carlo framework
to accommodate for the spin of the quarks in the jet,
and polar angle of the produced hadrons’ transverse momenta with respect to the direction of the initial quark’s
spin. The polar-angle dependent elementary polarized
fragmentations functions, that are a sum of elementary
unpolarized and Collins functions, were used as inputs for
the MC simulations. Here the unpolarized function was
taken the one used earlier in NJL-jet model, while the
Collins function was taken from calculations within the
spectator model of [11], with the parameters and regularization of transverse momentum integrals of the NJL-jet
model. Using this input, we calculated the corresponding number densities for the produced hadrons in MC
simulations, where we fixed the number of the produced
hadrons in each quark-jet (NLinks ). Then, we used the
form of the polar angle dependence of the full number
densities of Eq. (3) to separate the unpolarized and the
Collins functions.
14
s h
2.0
0.6
π +, z=0.7
K +, z=0.7
K -, z=0.7
s h
0.4
π +, z=0.7
K +, z=0.7
K -, z=0.7
P⊥
H⊥
z mh 1
P⊥
H ⊥ / D1
z mh 1
1.5
0.2
1.0
0
0.5
−0.2
0
0
0.2
0.4
0.6
0.8
1.0
0
0.2
0.4
0.6
P⊥2 (GeV 2)
P⊥2 (GeV 2)
(a)
(a)
0.8
1.0
1.5
s h
s h
π +, z=0.1
K +, z=0.1
K -, z=0.1
π +, z=0.1
K +, z=0.1
K -, z=0.1
0.2
P⊥
H⊥
z mh 1
P⊥
H ⊥ / D1
z mh 1
1.0
0.5
0
0
−0.2
−0.5
0
0.2
0.4
0.6
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
P⊥2 (GeV 2)
P⊥2 (GeV 2)
(b)
(b)
FIG. 21. The Collins fragmentation function vs P⊥2 for pions
and kaons produced by a s quark with z = 0.7 (a) and z = 0.1
(b) for NLinks = 6.
FIG. 22. The ratio of Collins fragmentation function to unpolarized fragmentation vs P⊥2 for pions and kaons produced
by a s quark with z = 0.7 (a) and z = 0.1 (b) for NLinks = 6.
Results for two scenarios were presented. First, we
showed the results for P⊥2 integrated case using a toy
model, where we assumed the form of the elementary
Collins term to be that of the unpolarized one times
0.1. Also, the simulations were performed with only light
quarks and pions for simplicity. The results for the so⊥(1/2)
lutions of H1
(z) showed that a nonzero unfavored
Collins function can be generated in our model, with
a sign opposite to that of the favored one. Also, both
the unfavored and favored Collins functions oscillate for
the lower values of z. Second, we presented the results
with the full model calculations of both unpolarized and
Collins functions with light and strange quarks, as well as
pions and kaons. The resulting full solutions of the 1/2
moment of the Collins functions exhibit features similar to the ones from the toy model. Interestingly, for
the solutions for the kaons produced by a u quark, the
magnitudes of the unfavored solutions were only slightly
smaller than the favored ones.
Finally, we studied the P⊥2 dependence of the Collins
function for several fixed values of z. Here we noticed
that the multiple hadron emissions produce modulations
of the polarized number densities, which contain several
powers of sin(ϕ). These higher order modulations are
a genuine feature of the quark-jet hadronization mechanism and neither depend on the particular forms for the
elementary fragmentation functions nor are they artifacts
of our Monte Carlo simulations. A detailed study, which
will be presented in our forthcoming publications [29, 30],
show that this effect can be completely described by the
azimuthal modulation of the transverse momentum distribution of the remnant quark in each elementary fragmentation process and to the azimuthal modulations arising from the quark spin flip probability. The fitting functions for obtaining the unpolarized and Collins functions,
described in Sec. IV, were modified using Eq. (29) to account for this effect, which shows up only in the unintegrated TMD fragmentation functions. Our results show
that the Gaussian function cannot be used to reliably
model the Collins functions in any regions of z.
15
We also investigated the Schäfer-Teryaev sum rule for
our results, and found that the naı̈ve sum rule of Ref. [31]
cannot be satisfied with our solutions, even though the
transverse momentum conservation is explicitly enforced
in our MC framework. The omission of the remnant
quark transverse momentum is the key here, as it was
proven explicitly from the definition of the Collins function in Ref. [32] that the sum rule holds if the quarkto-quark Collins functions are also accounted for. We
argued that the remnant quark in the hadronization process after several hadron emissions, though carrying a
minuscule fraction of the initial quark’s light-cone momentum, acquires a significant transverse momentum on
average, thus needs to be included in this sum rule that
manifests the transverse momentum conservation in the
hadronization process.
Our results for Collins functions have some distinctive
features that were approximately observed in the experiment: the similar size and the opposite sign for the 1/2
moments of favored and unfavored ones. At this stage,
however, a direct comparison with the experimental data
is not possible because we did not perform the QCD evolution on our results, as these equations for the Collins
function are not yet known. The recent work on the
evolution of TMD distribution and fragmentation functions of Ref. [33] pave the way for this, and we plan to
implement the QCD evolution of Collins functions in our
forthcoming work as the relevant formalism is developed.
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ACKNOWLEDGEMENTS
This work was supported by the Australian Research
Council through Grants No. FL0992247 (AWT), No.
CE110001004 (CoEPP), and by the University of Adelaide.